Homework #2
Math 6070-1, Spring 2006
Let X1 , X2 , . . . , Xn be random variables, all i.i.d., and with the same distribution function F that has density f := F 0 . Define for all p ≥ 1,
Q(p)
n (F )
Z
∞
:=
−∞
1/p
p
.
F̂n (x) − F (x) f (x) dx
This is a kind of “distance” between F̂n and F , although it is different from
Dn (F ).
(p)
(p)
1. Prove that Qn (F ) ≤ 2, so Qn (F ) is always finite.
(p)
2. Compute Qn (Fu ) where Fu (x) = x for 0 ≤ x ≤ 1, Fu (x) = 0 if x < 0,
and Fu (x) = 1 if x ≥ 1.
(p)
3. Prove that Qn (F ) is distribution-free. That is, if F and G are strictly
increasing distribution functions such that F 0 and G0 both exist, then the
(p)
(p)
distribution of Qn (F ) is the same as that of Qn (G).
P
(p)
4. Prove that under F , as n → ∞, Qn (F ) → 0, provided that F is strictly
increasing and has a density. Do this by first proving that
2 1
(2)
EF Qn (F ) =
.
6n
5. Suppose F is strictly increasing has a density. Then provide a heuristic
justification of the fact that, under F , as n → ∞,
√
d
n Qn(p) (F ) →
Z
1
1/p
p
|B ◦ (x)| dx
,
0
◦
where B denotes the Brownian bridge on [0 , 1]. Later on in a Project we
will see how to simulate the distribution of the latter limiting object.
6. Use 4 to test H0 : F = F0 versus H1 : F 6= F0 for a known distribution
function F0 that is strictly increasing and has a density.
7. Suppose F has a density f which satisfies f (x) > 0 for all x. Then prove
that F is strictly increasing.